# On product affine hyperspheres in $\mathbb{R}^{n+1}$

## TL;DR

This paper classifies locally strongly convex affine hyperspheres in Euclidean space that are locally isometric to products of constant curvature manifolds, including special cases with parallel Ricci tensor in dimensions 3 and 4.

## Contribution

It provides a complete classification of certain affine hyperspheres with specific geometric structures, extending understanding of their global geometry.

## Key findings

- Classification of affine hyperspheres as Riemannian products of constant curvature manifolds.
- Explicit descriptions of hyperspheres with parallel Ricci tensor in dimensions 3 and 4.

## Abstract

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space $\mathbb{R}^{n+1}$ which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvatures. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, affine hyperspheres of dimensions 3 and 4 with parallel Ricci tensor are also classified.